Adaptive, anisotropic and hierarchical cones of discrete convex functions

Abstract

We introduce a new class of adaptive methods for optimization problems posed on the cone of convex functions. Among the various mathematical problems which possess such a formulation, the Monopolist problem (Rochet and Choné, Econometrica 66:783–826, 1998; Ekeland and Moreno-Bromberg, Numer Math 115:45–69, 2010) arising in economics is our main motivation. Consider a two dimensional domain \(\Omega \), sampled on a grid \(X\) of N points. We show that the cone \({{\mathrm{Conv}}}(X)\) of restrictions to \(X\) of convex functions on \(\Omega \) is typically characterized by \(\approx N^2\) linear inequalities; a direct computational use of this description therefore has a prohibitive complexity. We thus introduce a hierarchy of sub-cones \({{\mathrm{Conv}}}(\mathcal{V})\) of \({{\mathrm{Conv}}}(X)\), associated to stencils \(\mathcal{V}\) which can be adaptively, locally, and anisotropically refined. We show, using the arithmetic structure of the grid, that the trace \(U_{|X}\) of any convex function U on \(\Omega \) is contained in a cone \({{\mathrm{Conv}}}(\mathcal{V})\) defined by only \(\mathcal{O}( N \ln ^2 N)\) linear constraints, in average over grid orientations. Numerical experiments for the Monopolist problem, based on adaptive stencil refinement strategies, show that the proposed method offers an unrivaled accuracy/complexity trade-off in comparison with existing methods. We also obtain, as a side product of our theory, a new average complexity result on edge flipping based mesh generation.